Properties

Label 1-140-140.47-r1-0-0
Degree $1$
Conductor $140$
Sign $0.581 + 0.813i$
Analytic cond. $15.0450$
Root an. cond. $15.0450$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s i·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s i·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(15.0450\)
Root analytic conductor: \(15.0450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (1:\ ),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.255214417 + 1.160396694i\)
\(L(\frac12)\) \(\approx\) \(2.255214417 + 1.160396694i\)
\(L(1)\) \(\approx\) \(1.495311715 + 0.3883604780i\)
\(L(1)\) \(\approx\) \(1.495311715 + 0.3883604780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.85499520739882166405739857171, −26.99959778898130180713556041994, −25.70739422534221020695790978853, −25.2434136811981980230924833257, −24.192699469359491657414123515586, −23.12795372130633747172369079840, −22.056004892239609430946602316669, −20.611187128011321491989587507603, −20.13960470977256529847415152266, −18.98330966401220851292678989407, −18.05548244122135985493511609440, −17.01448031049225724399290084404, −15.41868282256486643204366413867, −14.77930531361794192609790580319, −13.55260168198889383317949756145, −12.70640352357937553350317795505, −11.56995288879767092591011334315, −9.93974348343016951960079798727, −9.08323272661118704034430934011, −7.73858665759688174745846507369, −6.98661891481067929047546245050, −5.353037384564576941856486671555, −3.73732609196479043171494260049, −2.53794783760903725117212290033, −1.0231762246715073583886723454, 1.54492013853097838910467573472, 3.16336359287306547456603572788, 4.14232418989998477138201523338, 5.64736020855343388228397059197, 7.189276553850560130118894463641, 8.43865487461016943637248963266, 9.29700987341455081320940571443, 10.43481225475748593796296511272, 11.642477211966216687033990928833, 13.04905024969734540026523115717, 14.21026093020780456973071061444, 14.764472837316160555349378674187, 16.24148425486355992660296820658, 16.77696022478718778667880185410, 18.61591540371133657769757670940, 19.224356130609751863249244484457, 20.37075900772732898371436563922, 21.312931481905460106326095387037, 22.03034848633918089080289366821, 23.4004698991376074252699861677, 24.56910490768334234090826488679, 25.34424670996564052215053967120, 26.48515617010034821733197874499, 27.05497761400928320560777513942, 28.14982250827850056901160550654

Graph of the $Z$-function along the critical line